(a) Explanation Given information: ∫ 1 3 ( 2 x − 1 ) d x Formula: Δ x = b − a n T = Δ x 2 ( y 0 + y 1 ) | E T | = ∫ 1 3 ( 2 x − 1 ) d x − T Calculation: Show the actual value of definite integral as below: ∫ 1 3 ( 2 x − 1 ) d x (1) Show the definite integral f ( x ) as below: ∫ a b f ( x ) d x (2) Differentiate equation f ( x ) = 2 x − 1 with respect to x. f ' ( x ) = 2 Again differentiate the above equation with respect to x. f ' ' ( x ) = 0 The value of | f ' ' | is equal to 0. The term f ' ' is continuous and M is an upper bound for the values of | f ' ' | on [ a , b ] . Thus, the value of M becomes 0. Thus, it is recommended to estimate the minimum number of sub intervals by means of trial and error method. Calculate the exact value of the integral as given below. ∫ 1 3 ( 2 x − 1 ) d x = ( 2 x 2 2 − x ) 1 3 = 3 2 − 3 − ( 1 2 − 1 ) = 9 − 3 − 1 + 1 = 6 Compare equation (1) and (2). f ( x ) = 2 x − 1 (3) Substitute 1 for x (b) To determine The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10 − 4 by Simpson’s rule.

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

4th Edition

ISBN: 9780136880912

Author: Joel Hass, Christopher Heil

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 8.6, Problem 12E

(a)

To determine

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Trapezoidal rule.

(b)

To determine

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Simpson’s rule.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.6, Problem 11E

Chapter 8.6, Problem 13E

Students have asked these similar questions

This question builds on an earlier problem. The randomized numbers may have changed, but have your work for the previous problem available to help with this one. A 4-centimeter rod is attached at one end to a point A rotating counterclockwise on a wheel of radius 2 cm. The other end B is free to move back and forth along a horizontal bar that goes through the center of the wheel. At time t=0 the rod is situated as in the diagram at the left below. The wheel rotates counterclockwise at 1.5 rev/sec. At some point, the rod will be tangent to the circle as shown in the third picture. A B A B at some instant, the piston will be tangent to the circle (a) Express the x and y coordinates of point A as functions of t: x= 2 cos(3πt) and y= 2 sin(3t) (b) Write a formula for the slope of the tangent line to the circle at the point A at time t seconds: -cot(3πt) sin(3лt) (c) Express the x-coordinate of the right end of the rod at point B as a function of t: 2 cos(3πt) +411- 4 -2 sin (3лt) (d)…

5. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.5.AE.003. y y= ex² 0 Video Example x EXAMPLE 3 (a) Use the Midpoint Rule with n = 10 to approximate the integral कर L'ex² dx. (b) Give an upper bound for the error involved in this approximation. SOLUTION 8+2 1 L'ex² d (a) Since a = 0, b = 1, and n = 10, the Midpoint Rule gives the following. (Round your answer to six decimal places.) dx Ax[f(0.05) + f(0.15) + ... + f(0.85) + f(0.95)] 0.1 [0.0025 +0.0225 + + e0.0625 + 0.1225 e0.3025 + e0.4225 + e0.2025 + + e0.5625 €0.7225 +0.9025] The figure illustrates this approximation. (b) Since f(x) = ex², we have f'(x) = 0 ≤ f'(x) = < 6e. ASK YOUR TEACHER and f'(x) = Also, since 0 ≤ x ≤ 1 we have x² ≤ and so Taking K = 6e, a = 0, b = 1, and n = 10 in the error estimate, we see that an upper bound for the error is as follows. (Round your final answer to five decimal places.) 6e(1)3 e 24( = ≈

2. [-/1 Points] DETAILS MY NOTES SESSCALCET2 6.5.015. Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) ASK YOUR TEACHER 3 1 3 + dy, n = 6 (a) the Trapezoidal Rule (b) the Midpoint Rule (c) Simpson's Rule Need Help? Read It Watch It

Chapter 8 Solutions

Pearson eText University Calculus: Early Transcendentals -- Instant Access (Pearson+)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10 − 4 by Trapezoidal rule.

Videos

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Trapezoidal rule.

The minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10−4 by Simpson’s rule.

Want to see the full answer?

Chapter 8 Solutions